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Quarter Comma Meantone

In the Meantone Temperament, the idea is to abandon the perfect fifths of the Pythagorean Temperament in favor of perfect major thirds.

For historical reasons the Meantone tuning starts at Eb. Now, in order to make the 4 fifth distance Eb-G a perfect major third, we have to multiply the fifth interval by

$\text{MF} = \sqrt[{\Large 4}\;]{\frac{5/4}{(3/2)^4/4}} = 0.9968991874808$

where the division by 4 in the denominator stems from the octave cropping.

$\text{In cents: MFC} = \log(\text{MF}) \cdot \frac{100}{\log(\sqrt[12]{2})} = -5.3765724$

Since the Pythagorean fifth in cents reads $$\log(3/2) \cdot 100/(\log(\sqrt[12]{2})) = 701.95500$$, the diminished fifth will thus read $$701.955 - 5.3765724 = 696.5784276$$

Recall a cent to be the 100th part of a halftone in the Equal Temperament on a logarithmic scale.

This gives us a series:

Note MT Ratio Cropping
by Octave
In Cents Compare: Cents
Equal Temp.
Eb 1 = 1.0000 1.0000 0.00 0.00
Bb (3/2)1 * (MF)1 = 1.4953 1.4953 696.58 700.00
F (3/2)2 * (MF)2 = 2.2360 1.1180 1393.16 1400.00
C (3/2)3 * (MF)3 = 3.3437 1.6719 2089.74 2100.00
G (3/2)4 * (MF)4 = 5.0000 1.2500 2786.31 2800.00
D (3/2)5 * (MF)5 = 7.4767 1.8692 3482.89 3500.00
A (3/2)6 * (MF)6 = 11.1803 1.3975 4179.47 4200.00
E (3/2)7 * (MF)7 = 16.7185 1.0449 4876.05 4900.00
B (3/2)8 * (MF)8 = 25.0000 1.5625 5572.63 5600.00
F# (3/2)9 * (MF)9 = 37.3837 1.1682 6269.21 6300.00
C# (3/2)10 * (MF)10 = 55.9017 1.7469 6965.78 7000.00
G# (3/2)11 * (MF)11 = 83.5925 1.3061 7662.36 7700.00
(D#) (3/2)12 * (MF)12 = 125.0000 1.9531 8358.94 8400.00

The D# does not exist in the Meantone Temperament, it is here to finish the circle. Because we have stacked slightly diminuished fifths, the meantone circle cannot close. With 12 meantone fifth we are 41.06 cents short! This interval has a name, it is called Diesis (some people say "lesser" diesis, because there are other definitions which slightly differ and give slightly different values, but these are not so important here, so I omit the "lesser").

The "evil" fifth G# - Eb, detuned by the diesis 41.06 cents, is called Wolf Fifth.

Note that the difference between the Meanatone fifth and the perfect fifth, calculated to 5.377 cents, is about the 4th part of the Pythagorean Comma 23.46 cents - this is where the name "Quarter Comma Meantone" comes from.

Also note that the major second (the F in the table above) with its frequency factor of 1.118 lies pretty good in the middle between the major second given by the Pythagorean tuning by building 2 perfect fifth above the Eb, which calculates to (3/2)2/2 = 1.118 + 0.007, and the other major second by building 10 perfect fifth above the Eb, which calculates to 1 / [ (3/2)10 / 64 ] = 1.118 - 0.008. This is where the mean in Meantone comes from.